Riemann Hypothesis From Area Quantization, Renormalization Group

On the Riemann Hypothesis, Area Quantization, Dirac Operators, Modularity and Renormalization Group Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, GA. 30314, [email protected] 8 December 2008, Revised December 17, 2008 Abstract Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert-Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line : sn = 21 + iρn . A detailed analysis of a one-dimensional Dirac-like operator with a potential V (x) is given that reproduces the spectrum of energy levels En = ρn , when the boundary conditions ΨE (x = −∞) = ± ΨE (x = +∞) are imposed. Such potential V (x) is derived implicitly from the relation x = x(V ) = π2 (d N (V )/dV ), where the functional form of N (V ) is given by the full-fledged Riemann-von Mangoldt counting function of the zeta zeros, including the f luctuating as well as the O(E −n ) terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr-Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes 1 )) has a one-to-one correspondence distribution for very large x (O( logx with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the Renormalization Group program.

Keywords: Quantum Mechanics, Dirac Operators, Riemann Hypothesis, HilbertPolya conjecture, Area Quantization, Modularity, Renormalization Group, String Theory.

1

1

Introduction : Riemann Hypothesis, Scaling and Modular Invariance

Riemann’s outstanding hypothesis [1] that the non-trivial complex zeros of the zeta-function ζ(s) must be of the form sn = 1/2 ± iρn , is one of most important open problems in pure mathematics. The zeta-function has a relation with the number of prime numbers less than a given quantity and the zeros of zeta are deeply connected with the distribution of primes [1]. References [2] are devoted to the mathematical properties of the zeta-function. The RH has also been studied from the point of view of mathematics and physics [6], [9], [14], [18], [11], [19] among many others. A novel physical interpretation of the location of the nontrivial Riemann zeta zeros which corresponds to the presence of tachyonic-resonances/tachyonic-condensates in bosonic string theory was found in [7] : if there were zeros outside the critical line violating the RH these zeros do not correspond to any poles of the string scattering amplitude. The spectral properties of the ρn ’s are associated with the random statistical fluctuations of the energy levels (quantum chaos) of a classical chaotic system [14]. Montgomery [8] has shown that the two-level correlation function of the distribution of the ρn ’s coincides with the expression obtained by Dyson with the help of random matrices corresponding to a Gaussian unitary ensemble. In [10] by constructing of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series and logarithmic derivatives, and after invoking a CT -symmetry corresponding to a judicious charge conjugation C and time reversal T operation, we were able to show that the Riemann Hypothesis follows. The charge conjugation operation C is related to scalings transformations, and time reversal T operation, is related to the inversions t → (1/t) such that log(t) → −log(t). A ” Wick rotation” of variables t = iz furnishes z → −(1/z) which is a modular SL(2, Z) transformation z → (az + b/cz + d) with unit determinant ad − bc = 1. For these reasons, before entering into the next two sections we deem it very important to review the results [10], [16] based on a family of scalinglike operators in one dimension involving the Gauss-Jacobi theta series and an infinite parameter family of theta series where the inner product of their eigenfunctions Ψs (t; l) is given by (2/l)Z [ 2l (2k − s∗ − s)], where Z(s) is the Riemann completed zeta function and the l, k parameters are constrained to obey (l + 4)/8 = k in order to have CT -invariance. There is a one-to-one correspondence among the zeta zeros sn ( Z[sn ] = 0 ⇒ ζ(sn ) = 0 ) with the eigenfunctions Ψsn (t; l) (of the latter scaling-like operators) when the latter are orthogonal to the ”ground” reference state Ψso (t; l); where so = 21 + i0 is the center of symmetry of the location of the nontrivial zeta zeros . We shall present a concise review [10] and show why the Riemann Hypothesis follows from a CT invariance when the pseudo-norm of the eigenfunctions < Ψs |CT |Ψs > is not null. Had the pseudo-norm < Ψs |CT |Ψs > been null, the RH would have been false.

2

The Scaling Operators related to the Gauss-Jacobi Theta series and the Riemann zeros [16] are given by d dV + + k. (1.1) d ln t d ln t such that its eigenvalues s are complex-valued, and its eigenfunctions are given by ψs (t) = t−s+k eV (t) . (1.2a) D1 = −

D1 is not self-adjoint since it is an operator that does not admit an adjoint extension to the whole real line characterized by the real variable t. The parameter k is also real-valued. The eigenvalues of D1 are complex valued numbers s. The charge conjugation operation C acting on the eigenfunctions is defined as ∗ ψs (t) = t−s+k eV (t) → ψs∗ (t) = t−s +k eV (t) = t−s

∗

+s

ψs (t).

(1.2b)

which is related to scalings transformations of ψs (t) by t-dependent (local) scaling factors t−s

∗

+s

= e(−s

∗

+s) ln t

=e

2 i Im(s) lnt

⇒ a phase rotation

(1.2c)

where Im(s) is the imaginary part of s. Since local t-dependent (lnt dependent to be precise ) phase rotations resemble U (1) gauge transformations one can then interpret the (dV /dlnt) term in D1 as a gauge field (potential) in onedimension that gauges the scalings transformations. V is the pre-potential and A = (dV /dlnt) is the potential. Thus charge conjugations (1.2b) can be recast as scaling transformations (1.2c). We also define the ”mirror” operator to D1 as follows, D2 =

dV (1/t) d − + k. d ln t d ln t

(1.3)

that is related to D1 by the substitution t → 1/t and by noticing that dV (1/t) dV (1/t) =− . d ln(1/t) d ln t

(1.4)

where V (1/t) is not equal to V (t) and D2 is not self-adjoint either. When l = 4(2k − 1), the eigenfunctions of the D2 operator are Ψs ( 1t ) (with eigenvalue s), and which can be shown to be equal to Ψ1−s (t) [16]. This results from the properties of the Gauss-Jacobi theta series under the x → 1/x transformations. Since V (t) can be chosen arbitrarily, we chose it to be related to the Bernoulli string spectral counting function, given by the Jacobi theta series, 2V (t)

e

=

∞ X

e−πn

n=−∞

3

2 l

t

= 2ω(tl ) + 1.

(1.5)

this is where the l parameter appears in (1.5); the k parameter appears in (1.1, 1.3). The condition l = 4(2k − 1) [16] is required so the orthogonal states Ψsn (t) (parametrized by the complex eigenvalues sn ) to the ground state Ψs=1/2 (t) have a one-to-one correspondence to the zeta zeros zn in such a way that the quartets of numbers {sn } are symmetrically located w.r.t the critical line, and real axis, in the same way that the zeta zeros zn are : the quartets are {sn } = sn ; 1 − sn ; s∗n ; 1 − s∗n . Furthermore the condition l = 4(2k − 1) is required in order construct CT -invariant (but not Hermitian) Hamiltonians as we describe below. The related theta function defined by Gauss was G(1/x) =

∞ X

e−πn

2

/x

= 2ω(1/x) + 1.

(1.6)

n=−∞

where ω(x) =

∞ P

2

e−πn x . The Gauss-Jacobi series obeys the relation

n=1

√ 1 (1.7) G( ) = x G(x). x resulting from the Poisson re-summation formula. The V (t) is defined as e2V (t) = G(tl ) where x = tl . The pair of mirror Hamiltonians HA = D2 D1 and HB = D1 D2 , when l = 4(2k − 1) obey HA Ψs (t) = s(1 − s)Ψs (t).

1 1 HB Ψs ( ) = s(1 − s)Ψs ( ). t t

(1.8)

due to the relation Ψs (1/t) =√Ψ1−s (t) based on the modular properties of the Gauss-Jacobi series, G( x1 ) = x G(x). Therefore, despite that HA , HB are not Hermitian they have the same spectrum s(1 − s) which is real-valued only in the critical line and in the real line. Eq-(1.8) is the one-dimensional version of the eigenfunctions of the two-dimensional hyperbolic Laplacian given in terms of the Eisenstein’s series. Had HA , HB been Hermitian one would have had an immediate proof of the RH. Hermitian operators have a real spectrum, hence if s(1 − s) is real this means that s = 21 + iρ, and/or s = real. The trivial zeta zeros are located at the negative even integers (real) and the nontrivial zeta zeros are located in the critical line s = 12 + iρ.√ From eq-(1.8) and using the properties of the Gauss-Jacobi series G( x1 ) = x G(x) it follows that under the ”time reversal ” T operation t → 1t the eigenfunctions Ψs (t) behave as 1 T Ψs (t) = Ψs ( ) = Ψ1−s (t). t

(1.9)

such that the Hamiltonian operators HA = D2 D1 , HB = D1 D2 transform as T HB T −1 = HA ,

T HA T −1 = HB .

(1.10a)

the combined action of CT transformations is implemented on the states as follows C T HA [ C T ]−1 Ψs (t) = HA Ψs (t) 4

C T HB [ C T ]−1 Ψs (t) = HB Ψs (t).

(1.10b)

since Ψs (t) span a continuum of eigenfunctions, for a continuum of s values, eqs(1.10b) result in the vanishing of the commutators [HA , CT ] = [HB , CT ] = 0. When the operators HA , HB commute with CT , there exits new eigenfunctions ∗ ∗ ΨCT s (t) of the HA operator with eigenvalues s (1 − s ). Let focus only in the HA operator since similar results follow for the HB operator. Defining | ΨCT s (t) > ≡ CT | Ψs (t) > .

(1.11) ∗

one can see that it is also an eigenfunction of HA with eigenvalue s (1 − s∗ ) : HA | ΨCT s (t) > = HA CT | Ψs (t) > = HA | Ψ1−s∗ (t) > = s∗ (1 − s∗ ) | Ψ1−s∗ (t) > = s∗ (1 − s∗ ) CT | Ψs (t) > = (Es )∗ | ΨCT s (t) > . (1.12) where we have defined (Es )∗ = s∗ (1 − s∗ ). The CT action on s(1 − s) Ψs is defined to be linear : s(1 − s) CT Ψs since C acts only on the states Ψs (t) as scalings (1.2b), and not on the numbers s(1 − s). Therefore, one has [HA , CT ] = 0 ⇒ < Ψs | [HA , CT ] | Ψs > = 0 ⇒ < Ψs | HA CT | Ψs > − < Ψs | CT HA | Ψs > = (Es )∗ < Ψs | CT | Ψs > − Es < Ψs | CT | Ψs > = (Es∗ − Es ) < Ψs | CT | Ψs > = 0.

(1.13)

Similar results follow for the HB operator. From (1.13) one has two cases to consider. • Case A : If the pseudo-norm is null < Ψs | CT | Ψs > = 0 ⇒ (Es − Es∗ ) 6= 0

(1.14)

then the complex eigenvalues Es = s(1 − s) and Es∗ = s∗ (1 − s∗ ) are complex conjugates of each other. In this case the RH would be false and there are quartets of non-trivial Riemann zeta zeros given by sn , 1 − sn , s∗n , 1 − s∗n . • Case B : If the pseudo-norm is not null : < Ψs | CT | Ψs > 6= 0 ⇒ (Es − Es∗ ) = 0 Es∗

(1.15) ∗

∗

then the eigenvalues are real given by Es = s(1 − s) = = s (1 − s ) and which implies that s = real ( location of the trivial zeta zeros ) and/or s = 1 2 + iρ ( location of the non-trivial zeta zeros). In this case the RH would be true and the non-trivial Riemann zeta zeros are given by sn = 12 + iρn and 1 − sn = s∗n = 12 − iρn . We are going to prove next why Case A does and cannot occur, therefore the RH is true because we are left with case B.

5

The inner product are defined as follows, Z∞ hf |gi =

f ∗g

dt . t

0

Based on this definition the inner product of two eigenfunctions of D1 is Z∞ hψs1 |ψs2 i =

e2V t−s12 +2k−1 dt

  . 2 2 = Z (2k − s12 ) , l l 0

(1.16)

where we have denoted s12 = s∗1 + s2 = x1 + x2 + i(y2 − y1 ) and used the expressions for the Gauss-Jacobi theta function and the definition of the completed zeta function Z[s] resulting from the Mellin transform as shown below. We notice that hψs1 |ψs2 i = hψso |ψs i, . (1.17) thus, the inner product of ψs1 and ψs2 is equivalent to the inner product of ψso and ψs , where so = 1/2 + i0 and s = s12 − 1/2. The integral is evaluated by introducing a change of variables tl = x (which gives dt/t = (1/l)dx/x) and using the result provided by the Gauss-Jacobi Theta given in Karatsuba and Voronin’s book [2]. The completed function Z[s] in eq-(1.16) can be expressed in terms of the Jacobi theta series, ω(x) defined by eqs-(1.5, 1.6) as Z∞ X ∞

2

e−πn x xs/2−1 dx =

0 n=1

Z

∞

xs/2−1 ω(x)dx

.

0

=

1 + s(s − 1)

Z

∞

[xs/2−1 + x(1−s)/2−1 ] ω(x)dx

1

= Z(s) = Z(1 − s), (1.18) where the completed zeta function is s Z(s) ≡ π −s/2 Γ( ) ζ(s), . 2

(1.19)

which obeys the functional relation Z(s) = Z(1 − s), which is a self-duality relation [26]. In [10] we recurred to an inf inite family of HA , HB operators associated with an inf inite family of potentials Vjm (t) corresponding to an infinite family of theta series with the advantage that no regularization of the inner products is 6

jm necessary. Another salient feature is that the pseudo-norm < Ψjm s | CT |Ψs > is not null ( see below eq-(1.28) ) as result that the zeta function ζ(s) has no zeros at s = 21 − 2m, m = 1, 2, 3, ...., ∞. The relevance of the behavior of ζ( 21 − 2m) 6= 0, m = 1, 2, 3, ..., ∞ is that it automatically avoids looking at the behavior of zeta at s = 1/2. Armitage [3] has found a zeta function ζL (s) defined over the algebraic number field L that has a zero at s = 1/2 and presumably satisfies the RH . This finding would not be compatible with the result of eq(1.16) and which was based on a regularized inner product. Therefore, the well defined inner product where no regularization is needed leads to the result (see jm below eq-(1.28) ) < Ψjm > ∼ ζ( 21 − 2m) 6= 0, m = 1, 2, 3, ..., ∞ s | CT |Ψs and which is no longer in variance with the behaviour of the zeta function ζL (s) defined over the algebraic number field L and that has a zero at s = 1/2 [3]. Analogous results follow if we had defined a new family of potentials V2j (t) in terms of a weighted theta series Θ2j (t) and whose Mellin transform yields the infinite family of extended zeta functions of Keating [4] and their associated completed zeta functions as shown by Coffey [5]. The Hermite polynomials weighted theta series associated to 2j = even degree polynomials are defined by

e2V2j (t) = Θ2j (t) ≡

n=∞ X

√ 2 (8π)−j H2j (n 2πt) e−πn t .

(1.20)

n=−∞

and are related to the potentials V2j (t) which appear in the definitions of the differential operators (1.1, 1.2). The weighted theta series obeys the relation 1 (−1)j √ Θ2j ( ) = Θ2j (t). t t

(1.21)

Only when j = even in (1.21) one can implement CT invariance in the new family of Hamiltonians HA , HB associated with the potentials V2j (t) of (1.20) because HA Ψs (t) = s(1 − s)Ψ(t) and HB Ψs ( 1t ) = s(1 − s)Ψs ( 1t ) would only be valid when j = even as a result of the relations (1.1, 1.2, 1.3) and (1.20, 1.21). The Mellin transform based on the weighted Θ2j (t) [5] requires once again to extract the zero mode n = 0 contribution of Θ2j (t) (to regularize the divergent integrals) in order to arrive at Z

∞

1 s [ Θ2j (t) − (8π)−2j H2j (0) ] ts/2−1 dt = Pj (s) π −s/2 Γ( ) ζ(s), Re s > 0. 2 2 0 (1.22) in the definition of the (regularized) inner products of the eigenstates associated to the new potentials (1.20). The polynomial pre-factor in front of the completed Riemann zeta Z(s) = π −s/2 Γ( 2s ) ζ(s) is given in terms of a terminating Hypergeometric series [5] Pj (s) = (8π)−j (−1)j

(2j)! s 1 2 F1 (−j, ; ; 2). j! 2 2

7

(1.23)

The orthogonal states Ψsn (t) to the ground state Ψso (t) ( so = 21 + i0) will now be enlarged to include the nontrivial zeta zeros and the zeros of the polynomial Pj (s). The polynomial Pj (s) has simple zeros on the critical line Re s = 12 , obeys the functional relation Pj (s) = (−1)j Pj (1 − s) and in particular Pj (s = 12 ) = 0 when j = odd [5]. It is only when j = even that Pj (s = 12 ) 6= 0 and when we can implement CT invariance resulting from the relation (1.21) and which is consistent with the results of eqs-(1.8, 1.10a. 1.10b). In order to avoid the regularization of the integrals involving the Mellin transform (1.22), we proposed another family of theta series where no regularization is needed in the construction of the inner products. There is a twoparameter family of theta series Θ2j,2m (t) that yield well defined inner products without the need to extract the zero mode n = 0 divergent contribution. Given

e2V2j,2m (t) = Θ2j,2m (t) ≡

n=∞ X

√ 2 n2m H2j (n 2πt) e−πn t .

(1.24)

n=−∞

when m 6= 0, the zero mode n = 0 does not contribute to the sum and the Mellin transform of Θ2j,2m (t) , after exploiting the symmetry of the even-degree Hermite polynomials, is [4], [5] Z

∞

[2 0

n=∞ X

√ 2 n2m H2j (n 2πt) e−πn t ] ts/2−1 dt =

n=1

s Γ( ) ζ(s − 2m); Re s > 1 + 2m, m = 1, 2, .... (1.25) 2 In order to find the analytical continuation of the Mellin transform (1.25) for all values of s in the complex plane we must use the analytical continuation of ζ(s) was found by Remann in his celebrated paper. A Poisson re-summation formula for Θ2j,2m (t) (1.24) leads to similar modular behaviour as eq-(1.21) and only when j = even one can implement CT invariance in the new family of Hamiltonians HA , HB associated with the new potentials V2j,2m (t) of (1.24) Therefore one has now at our disposal a well defined inner product of the states Ψs (t) (without the need to regularize it by extracting out the zero n = 0 mode of the theta series). In particular the inner product of the states Ψs (t) with the shif ted ”ground” states Ψ 21 +2m (t), m = 1, 2, .... corresponding to the potentials in (1.24), by recurring to the result (1.25) and following similar steps as in (1.16) is j

2 (8π) Pj (s) π

−s/2

s + 2m ) ζ(s). 2 (1.26) this result requires f ixing uniquely the values l = −2; k = 41 . The nontrivial zeta zeros sn correspond to the states Ψsn (t) orthogonal to the shifted ”ground” states Ψ 21 +2m (t) in eq-(1.26) : < Ψ 21 +2m (t) | Ψs (t) > = − 2 (8π)j Pj (s + 2m) π −(s+2m)/2 Γ(

8

< Ψ 12 +2m (t) | Ψsn (t) > = − 2 (8π)j Pj (sn + 2m) π −(sn +2m)/2 Γ( It remains to prove when l = −2, k = that

sn + 2m ) ζ(sn ) = 0; 2 1 4

m = 1, 2, 3, .....

(1.27) and s12 = s∗1 + s2 = s∗1 + (1 − s∗1 ) = 1

< Ψs | CT | Ψs > = < Ψs || Ψ1−s∗ > = Z

∞

[2 0

n=∞ X

√ 2(−s12 +2k) 2 −1 2l dt = n2m H2j (n 2πt) e−πn t ] t

n=1

1 1 1 − 2 (8π)j Pj (s = ) π −1/4 Γ( ) ζ( − 2m) 6= 0; 2 4 2

j = even, m = 1, 2, 3, .....

(1.28) Hence, one arrives at a definite solid conclusion based on a well defined inner product : because ζ( 21 − 2m) 6= 0 when m = 1, 2, ...., and Pj ( 12 ) 6= 0 when j = even in eq-(1.28), the pseudo-norm < Ψs | CT | Ψs > 6= 0, and this rules out case A in eq-(1.14) , and singles out case B in eq-(1.15) leading to the conclusion that Es = s(1 − s) = real ⇒ s = 12 + iρ ( and/or s = real ), and consequently the RH follows if, and only if, CT invariance holds. The key reason why the Riemann hypothesis follows is due to the CT invariance of the Hamiltonians HA , HB and that the pseudo-norm < Ψs |CT |Ψs > is not null. Had the pseudo-norm < Ψs |CT |Ψs > been null, the RH would have been false. It remains to be seen whether our procedure is valid to prove the grand-Riemann Hypothesis associated to the L-functions.

2

The Dirac and Schroedinger Operators that reproduce the zeta zeroes

The previous section was devoted to a family of scaling operators needed in the construction of a pair of non-Hermitian Hamiltonians, involving Θ(t) functions, whose spectrum Es = s(1 − s) = Es∗ = s∗ (1 − s∗ ) was shown to be real valued resulting from CT invariance, and whose solutions for s are s = 12 + iρ and/or s = real, showing how the Riemann Hypothesis is a physical realization of CT invariant QM. In this section we will find the Dirac-like Operator (with a potential V (x)) in one-dimension whose spectrum reproduces exactly the imaginary parts (ordinates) ρn = En of the zeta zeroes : ζ( 21 ± iEn ) = 0. At the end we will also provide a dif f erent potential V (x) associated with a Schroedinger operator in one-dimension that provides the same spectrum ρn = En The Dirac-like equation in one-dimension in the presence of a potential V (x) is

9

∂ + V (x) } ΨE (x) = E ΨE (x); ¯h = c = 1. (2.1) ∂x where the one-dim Clifford algebra with 21 = 2 generators is realized in terms of the unit element 1 and a 1 × 1 matrix γ whose entry is −i. Eq-(2.1) is the operator representation of the (constraint) dispersion relation { −i

∂ (2.2) ∂x such that the Bohr-Sommerfield quantization condition yields the number of energy levels E1 , E2 , ..., En (the number of the first n zeta zeroes on the critical line) Z xn Z xn 2 2 P(x) dx = [ En − V (x) ] dx = π xo =0 π xo =0 Z En 2 dx (En − V ) dV = N (En ) − N (Vo ). (2.3) π Vo dV P(x) + V (x) = E, when P → − i

We have set the lower integration limits at x = 0 because we assume that the potential is symmmetric. In fact, the potential will also turn out to be multivalued. The potential used by Wu-Sprung [13] for the Schroedinger operator ¯h2 ∂ 2 + VW S (x) } Ψ = E Ψ; ¯h2 = 2m = 1. (2.4) 2m ∂x2 turned out to be symmetric VW S (−x) = VW S (x) for the choice of the average energy level counting function given by { −

E 7 E [ log( ) − 1 ] + . (2.5) 2π 2π 8 (where log is the natural Neper logarithm in the Euler number base) after recurring to the solutions to Abel’s integral equation of the first kind obtained after differentiation w.r.t the E parameter of the Bohr-Sommerfield quantization condition N (E) =

2 π

Z

E

√

E − V dx =

0

2 π

Z

E

√

E−V

Vo

dx dV = N (E) − N (Vo ). dV

(2.6)

where Vo = V (x = 0) was chosen to obey the boundary condition N (Vo ) = 0

N (Vo ) =

Vo Vo 7 [ log( ) − 1 ] + = 0 ⇒ Vo ∼ 3.10073 π. 2π 2π 8

A differentiation of eq-(2.6) w.r.t to E (using Liebnitz rule) gives Z E Z E √ 2 d dx 1 1 dx √ E−V dV = dV = π dE Vo dV π Vo E − V dV 10

(2.7)

dN (E) 1 E = log( ). (2.8) dE 2π 2π The above equation belongs to the family of Abel’s integral equations associated with the unknown function f (V ) ≡ (dx/dV )

Jα [

dx 1 ] = dV Γ(α)

Z 0

E

(dx/dV ) 1 E dV = log( ); 1−α (E − V ) 2π 2π

0 < α < 1 (2.9)

The reason one had to differentiate eq-(2.6) w.r.t E is to enforce the condition 0 < α < 1 . Abel’s integral equation is basically the action of a fractional derivative operator J α [12] , for the particular value α = 12 , on the unknown function f (V ) = (dx/dV ) . Inverting the action of the fractional derivative operator (fractional anti-derivative) yields the solution for 1 d dx = dV Γ(1 − α) dV

Z 0

V

1 E 1 log ( ) dE. 2π 2π (V − E)α

(2.10a)

After setting the value α = 21 in (2.10a) , Wu-Sprung [13] found the solution in terms of quadratures for the dx/dV function, and a subsequent integration w.r.t V , gives √ √ 1 p Vo V + V − Vo 1 √ x = x(V ) = V − Vo log ( V log [ √ )+ ]. (2.10b) √ π 2πe2 π V − V − Vo where e = 2.71828... is Euler’s number and when V = Vo ⇒ xo = x(Vo ) = 0 consistent with the condition V (x = 0) = Vo ∼ 3.10073 π. The sought after potential VW S (x) that reproduces the average level density of zeta zeroes (energy eigenvalues) given by N (E) is implicitly given from the relation x(V ) upon inverting the function. It was after the fitting process of the first 500 Riemann zeta zeroes on the critical line when Wu-Sprung found numerically that a f ractal − shaped potential (obtained as a perturbation of the smooth VW S (x)) of dimension d = 1.5 was needed. A further fitting of the first 4000 zeroes furnished identical results for the fractal dimension d = 1.5 [15] associated with the shape of the potential. Based on these findings we proposed within the context of Supersymmetric QM a Weierstrass fractal function [16] as the fractal-shaped corrections to the smooth potential (2.10b) and consistent with the numerical findings by [13], [15] in order to model the fractal behaviour of the potential that fitted those zeta zeroes. Later on, Slater [17] performed an exhaustive detailed numerical analysis of our Weierstrass fractal function (and other fractal functions) to find a numerical fit for the first n = 25, 50, 75, .... zeta zeroes. The relevant feature of the expression for x(V ) (2.10b) is that it is explicitly given in terms of square roots (quadratures), such that changing the signs√of the square √ roots containing the variable V will yield a change of sign : x(− V ) = −x( V ) consistent with the assumption that V was symmetric V (−x) = V (x). 11

One can verify why this must be so from Abel’s solution (2/10a) . If (dx/dV ) changes to −(dx/dV ) by√replacing x → −x, leaving V fixed, one must take the minus sign of the 1/ V − E terms appearing in the r.h.s of (2.10a) when α = 12 . In general, when one replaces the average level counting function N (E) by the more general expression N (E) obtained by Riemann-von Mangoldt formula (given below in eq-(2.12)) one has the solution ( for α = 21 )

x(V ) − x(Vo ) =

1 Γ(1 − α)

Z

V

Z

dx 1 d = dV Γ(1 − α) dV

0 V

0

dN (E) 1 dE ⇒ dE (V − E)α

1 dN (E) 1 dE; α = . dE (V − E)α 2

(2.11)

At the end of this section we shall return to the solution (2.11) corresponding to the Schroedinger operator. Solutions to eq-(2.11) for a truncated version of the Riemann-von Mangoldt formula (2.12), where the oscillatory terms and the integral terms were dropped, was given by Slater [17] using the Mathematica Integrator package. By recurring to a Dirac-like operator it allows to use the f ull expression for number of zeroes (energy levels) N (E), including the fluctuating/oscillatory terms, leading to a symmetric and multi-valued potential due to the oscillatory terms in N (E). The coordinate function x(V ) is assumed to be single-valued, but its inverse, the potential V (x) is not necessarily single-valued, and in fact, it will turn out to be multi-valued. The typical example is the sine function x = sin(V ) (single-valued) whose inverse V = arcsin(x) is multi-valued. A knowledge of the functional form of the number of zeroes N (E) in the above integral-differential equation (2.3) gives the potential V (x) implicitly. Let us write the functional form for N (E) to be given by the Riemann-von Mangoldt formula which is valid for E ≥ 1

NRvM (E) =

E E 7 1 1 1 [ log( ) −1 ] + + arg [ ζ( +iE) ] + δ(E). (2.12) 2π 2π 8 π 2 π

where the (infinitely many times) strongly oscillating function is given by the argument of the zeta function evaluated in the critical line

S(E) =

1 1 1 1 arg [ ζ( + iE) ] = lim→0 Im log [ ζ( + iE + ) ]. (2.13) π 2 π 2

the argument of ζ( 12 + iE) is obtained by the continuous extension of arg ζ(s) along the broken line starting at the point s = 2 + i 0 and then going to the point s = 2 + iE and then to s = 21 + iE. If E coincides with the imaginary part of a zeta zero, then S(En ) = lim→0

1 [S(En + ) + S(En − )]. 2 12

(2.14)

An extensive analysis of the behaviour of S(E) can be found in [20]. In particular the property that S(E) is a piecewise smooth function with discontinuities at the ordinates En of the complex zeroes of ζ(sn = 21 + iEn ) = 0. When E passes through a point of discontinuity, En , the function S(E) makes a jump equal to the sum of multiplicities of the zeta zeroes at that point. The zeroes found so far in the critical line are simple [22]. In every interval of continuity (E, E 0 ), where En < E < E 0 < En+1 , S(E) is monotonically decreasing with derivatives given by S 0 (E) = −

E 1 log ( ) + O(E −2 ); 2π 2π

S 00 (E) = −

1 + O(E −3 ). (2.15) 2πE

The most salient feature of these properties is that the derivative S 0 (E) blows up at the location of the zeta zeroes En due the discontinuity (jump) of S(E) at En . Also, the strongly oscillatory behaviour of S(E) forces the potential V (x) to be a multi-valued function of x. The expression for δ(E) is [20] 1 1 E 1 E ) + arctan ( log (1 + ) − δ(E) = 4 4E 2 4 2E 2

Z

∞

ρ(u) du . 2 + (E/2)2 (u + 1/4) 0 (2.16) with ρ(u) = 21 − {u}, where {u} is the fractional part of u and which can be written as u − [u], where [u] is the integer part of u. In this way one can perform the integral involving [u] in the numerator by partitioning the [0, ∞] interval in intervals of unit length : [0, 1], [1, 2], [2, 3], .....[n, n + 1], .... The definite integral when the upper limit is bound by an ultra-violet regulator Λ is

−

E 2

Z 0

Λ

E (E/2)2 + (Λ + 1/4)2 ρ(u) du = log [ ] − 2 2 (u + 1/4) + (E/2) 4 (E/2)2 + (1/4)2 [Λ] X

n [ arctan(

n=1

4n + 5 4n + 1 ) − arctan( )] − 2E 2E

3 4Λ + 1 1 [ arctan( ) − arctan( )] 4 2E 2E It is the Euler-Maclaurin summation formula N −1 X k=1

Z fk =

N

f (k) dk 0

−

(2.17a)

1 [f (0) + f (N )] + 2

1 1 [f 0 (N ) − f 0 (0)] − [f 000 (N ) − f 000 (0)] + ..... (2.17b) 12 720 that permits the exact evaluation of the Λ → ∞ limit of the expression (2.17a): the divergent terms E log(Λ) in (2.17a) cancel out exactly leading to the δ(E) terms of (2.16) 13

E E2 5 1 + [ arctan( ) − arctan( )]+ 4 8 2E 2E 1 1 5 E 25 1 [ 33 arctan( ) −25 arctan( )]+ [ 5 log (1+ 2 ) − log (1+ 2 ) ] + 32 2E 2E 16 4E 4E 1 5 1 [ arctan( ) − arctan( ) ] + ........ (2.17c). 12 2E 2E For large E, a Taylor expansion of δ(E) gives δ(E) = −

1 1 1 (2.17d) + O( 3 ). 48 E E Thus the leading term of δ(E) is of the order (1/E) as expected. However, it is important to keep all the terms involving E given by (2.17c) when E is not large. We must search now for solutions to the integral equation associated with the Dirac-like operator in one-dimension δ(E) =

1 π

Z

E

( E−V ) Vo

dx dV = N (E) − N (Vo ) dV

(2.18)

dx associated with the unknown function f = f (V ) ≡ dV and subject to the boundary condition V (x = 0) = Vo . i.e. the integral transform of f (V ) defined by eq-(2.18) is the counting function N (E). The solutions to Abel’s integral equations will not be necessary in our case to find dx/dV . What should the choice of Vo be ? To answer this question we need to discuss the following points. The integral (2.18) is trivially zero when the upper limit E coincides with Vo , which is consistent with the trivial fact : N (E = Vo ) − N (Vo ) = 0. Despite that the Riemann-van Mangoldt expression (2.12) is only valid for E ≥ 1, one can still verify by inspection that when Vo = 0 ⇒ N (E = Vo = 0) = 0. This can be seen if one chooses the argument of ζ(1/2) = −1.46 to be given by −π, instead of π. With this choice for the argument and taking arctan(∞) = π2 , then (2.12) becomes

7 1 1 π + (−π) + = 0. (2.19) 8 π 4π 2 Had one chosen the argument π one would have N (E = 0) = 2 which is the wrong answer since there are no zeros at ζ(1/2). The choice Vo = 0 is a very natural one from the physical point of view and compatible with the E = 0 ground state of Supersymmetric Quantum Mechanics (SQM) in one-dimension. The super-potential W (x) in SQM vanishes at x = 0 if Supersymmetry is not broken. Upon taking derivatives on both sides of eq- (2.18 ) w.r.t to E gives N (E = 0) =

2 π

Z 0

E

dx(V ) 2 dV = dV π

Z

x

dx = 0

14

2x(E) = π

dN (E) dE

(2.20)

Notice that despite the derivatives N 0 (E) blow up at the location of the zeta zeroes E = En , due the discontinuity (jump) of S(E) at En , the expression (2.20) is nevertheless correct because it just means that the function x(E) also blows up x(En ) = ∞ when E = En . Therefore, the fact that N 0 (E) blows up at a discrete number of locations E = En does not preclude us from differentiating both sides of eq-(2.18) w.r.t to E. From the above relation (2.20) we will show that the solution to the integral equation (2.18) is 2x(V ) dN (V ) 2x = = ≡ ρ(V ). (2.21) π π dV where N (V ) has the same f unctional f orm as N (E). The physical interpretation of (2.21) is that the coordinate function x = x(V ) is just proportional to the density of zeroes ρ(V ) (times π/2) : the number of zeros per unit of energy. On dimensional grounds this makes sense, since length has the dimensions of inverse of energy when h ¯ = c = 1. Therefore, one can infer that when V = En ⇒ x = x(En ) = xn = ∞ for all values of n = 1, 2, 3, ..... due to the singular behaviour of the derivatives N 0 (V = En ) at the ordinates of the zeta zeroes resulting from the discontinuity of the argument of the zeta function at En . The last expression (2.21) for x(V ) furnishes the sought-after potential V = V (x) in implicit form. x(V ) is single-valued but V (x) is multi-valued. In particular, V (x = ∞) = En , n = 1, 2, 3, ...... Equipped with the known expression for the functional form of N (V ) (2.12) (after replacing E for V ) the quantization condition (2.18) reads Z

E

(E − V ) Vo

d2 N (V ) dV = N (E) − N (Vo ) . dV 2

(2.22)

Taking derivatives on both sides of (2.22) w.r.t to E and using the most general Liebnitz formula for differentiation of a definite integral when the upper b(E) and lower b(E) limits are functions of a parameter E : d dE

Z

b(E)

Z

b(E)

f (V ; E) dV = a(E)

( a(E)

f (V = b(E) ; E) ( leads to Z

∂f (V ; E) ) dV + ∂E

d b(E) d a(E) ) − f (V = a(E) ; E) ( ). dE dE E

Vo

d2 N (V ) dN (E) dV = . 2 dV dE

(2.23)

(2.24)

since the lower limit Vo is taken to be independent of E and the integrand vanishes in the upper limit V = b(E) = E. The integral (2.24) is straightforward dN (V ) dN (V ) dN (E) (V = E) − (Vo ) = . dV dV dE

15

(2.25)

from which one infers that one must have N 0 (Vo ) = dNdV(V ) (Vo ) = 0 since the f unctional forms of N (V ) and N (E) are the same. However, there is a potential problem because there is no assurance that the function N (V ) obeys the condition N 0 (Vo ) = 0 unless one chooses the value of Vo to be the solution to the transcendental equation i ζ 0 ( 21 + iVo ) 3 1 Vo 1 1 1 1 ) − − log ( ) + Im + log (1 + 1 2 2π 2π π 4π 4Vo 2π 1 + 4Vo2 ζ( 2 + iVo ) 1 2π

Z

∞

ρ(u) du Vo + 2 2 (u + 1/4) + (Vo /2) 2π

Z

∞

ρ(u) (Vo /2) du = 0. [ (u + 1/4)2 + (Vo /2)2 ]2 0 0 (2.26) If, and only if, there is a real-valued solution Vo to eq-(2.26) that fixes the value of the zero-point energy Vo and that falls in the range 1 ≤ Vo < E1 , then one has that x(V ) = π2 (dN (V )/dV ) yields the potential V (x) in implicit form reproducing the ordinates of the zeta zeros for the spectrum En . However, if there is no real-valued solution Vo to the transcendental equation (2.26) that falls in the range 1 ≤ Vo < E1 , then one can go ahead and truncate the upper limit of the definite integral appearing in the definition of δ(E) in eqs(2.16, 2.17) by introducing an E-dependent ultraviolet cutoff Λ = Λ(E), such that the zero derivative condition is modified from the form given by (2.26) to the one given by N 0 (Vo , Λ(Vo )) = i ζ 0 ( 21 + iVo ) 1 Vo 1 1 1 3 1 log ( ) + Im + log (1 + ) − − 2π 2π π 4π 4Vo2 2π 1 + 4Vo2 ζ( 12 + iVo ) Z Λ(Vo ) Z Λ(Vo ) 1 ρ(u) du Vo ρ(u) (Vo /2) du + − 2 2 2π 0 (u + 1/4) + (Vo /2) 2π 0 [ (u + 1/4)2 + (Vo /2)2 ]2 Vo ρ(Λ(Vo )) dΛ(V ) ( )(V = Vo ) = 0. 2π [ (Λ(Vo )) + (1/4) ]2 + (Vo /2)2 dV

(2.27)

Therefore, the above condition N 0 (Vo , Λ(Vo )) = 0 provides the necessary constraint between Vo and Λ(Vo ) to satisfy our goal. It is customary in the Renormalization Group process in Quantum Field Theories (QFT) to introduce an energy cut-off; here Λ(E) is a running and increasing function of E which tends to infinity when E → ∞. To sum up, if there is a real-valued solution Vo to eq-(2.26) that fixes the value of the zero-point energy Vo and that falls in the range 1 ≤ Vo < E1 , then x(V ) = π2 (dN (V )/dV ) yields the potential V (x) in implicit form. If there is no real-valued solution Vo to the transcendental equation (2.26) that falls in the range 1 ≤ Vo < E1 , then one truncates the upper limit of the definite integral )) determines the leading to a modified equation (2.27) and x(V ) = π2 dN (V,Λ(V dV potential implicitly in terms of the cut-off function Λ(V ).

16

Next we describe how one determines the functional form of the cut-off function Λ(V ) in such a case. Because Λ(E) is a cutoff-function that runs with energy E, one has now enough freedom to impose the exact conditions N (En ; Λ(En )) − N (Vo ; Λ(Vo )) = n;

n = 1, 2, 3, ..... ; 1 ≤ Vo < E1 (2.28a) when E1 , E2 , E3 , .......En , En+1 , ..... are the (positive) imaginary parts (ordinates) of the zeta zeroes in the critical line. In order to evaluate N (V ) at V = En due to the discontinuity of the fluctuating term S(E) of (2.12) at En one must take the arithmetic mean as described by eq-(2.14). The upper limit of the values of Vo should be bounded by the first zero E1 avoiding having potential singularities in eq-( 2.27) due to the zeros of zeta appearing in the denominator of second term. The lower bound of Vo should be 1 since the domain of validity of the Riemann-van Mangoldt expression is E ≥ 1. If one were to replace the values Λ(En ) = λn for Λ = ∞ one may rewrite eq-(2.28) as follows N (En ; λn ) −N (Vo ; Λ(Vo )) = [ N (En ; Λ = ∞) + ∆(λn , En ) ] − N (Vo ; Λ(Vo )) = (n − δn ) + ∆(λn , En ) − N (Vo ; Λ(Vo )) = n.

(2.28b)

where the number of levels (zeros) just below the n-th zero given by the Riemannvan Mongoldt expression are N (En ; Λ = ∞) = n − δn (if the Riemann Hypothesis is true), and δn is a fraction such that 0 < δn < 1. The positive-definite quantity ∆(λn , En ) is Rthe deficit value of the integral appearing in eqs-(2.16, ∞ 2.17) given by (En /2) λn (....). Finally, one can derive implicitly the potential of the Dirac-like operator that reproduces the zeta zeros from 2x(V ) dN (V ; Λ(V )) ∂N (V ; Λ(V )) dΛ(V ) ∂N (V ; Λ(V )) 2x = = = + ⇒ π π dV ∂V dV ∂Λ(V ) π dN (V ; Λ(V )) (V = Vo ) = 0. 2 dV Next we describe how to solve the system of eqs-(2.27-2.28). Firstly, one begins by truncating the series expansion for the cut-off function Λ(V ) as follows xo = x(Vo ) =

Λ(V ) =

k=∞ X

ak V k ⇒ Λ(V ; N ) =

k=0

(

k=N X

ak V k ⇒

k=0

k=N X dΛ(V, N ) ) = ak k V k−1 dV

(2.29)

k=0

We are going to display two case scenarios how to solve eqs-(2.27-2.28). In the first case we are going to drastically simplif y these equations by choosing 17

Vo = 0, despite that the domain of values for Vo in the definition of N (Vo ) given by the Riemann-van Mongoldt formula should be 1 ≤ Vo < E1 . In the second case scenario we shall enforce the latter conditions on Vo . Hence, eqs-(2.27-2.28) are automatically simplified by setting Vo = 0 ⇒ N (Vo = 0 ; λo ) = 0, which follows from eq-(2.19 ), so now the value of λo = Λ(Vo = 0) is determined from the relation (2.27) for Vo = 0 iζ 0 ( 12 ) 3 1 1 1 − − log (4π) + Im − 2π π 2π 2π ζ( 12 )

Z 0

λo

ρ(u) du = 0. (2.30) (u + 1/4)2

after noticing a cancellation between the singular log(Vo = 0) − log(Vo = 0) terms associated with the first and third terms of (2.27) resulting from the relation limVo →0 (1/4π) log(1 + 4V1 2 ) → −(1/2π) log(Vo = 0) = +∞. The value o of the integral in (2.30) can be inferred from (−2/E)× the value of the integral in eq-(2.17). When E → 0, it requires the use of L’Hopital’s rule giving a finite result for a finite value of λo . The values ζ 0 ( 21 ) = −3.92265; ζ( 12 ) = −1.46 in eq-(2.30) yields a negative value for λo given by −0.0787 =

1 2

Z 0

λo

ρ(u) du 1 = − (u + 1/4)2 2

Z

0

λo

ρ(u) du 1 ; with − < λo < 0. (u + 1/4)2 4

The quantization conditions (2.28a, 2.28b) corresponding to N (Vo = 0; λo ) = 0, and Λ(Vo = 0) ≡ λo < 0 given by the solution to eq-(2.30), and by writing Λ(En ) ≡ λn , become N (En ; λn ) − N (Vo = 0 ; λo ) = N (En ; λn ) = n; n = 1, 2, 3, ......... ∞ ⇒ Z ∞ En ρ(u) du ∆(λn , En ) = = δn = n − N (En ; Λ = ∞). 2 λn [ (u + 1/4)2 + (En /2)2 ] (2.31) ∆(λn , En ) is positive definite because the contributions to the integral in all of the intervals [ [u], [u] + 1 ] which do not contain λn are all positive due to the increasing values of the denominator, whereas the magnitude of the values ρ(u) = 21 −{u} are symmetrically distributed about the midpoint of the intervals while being positive and negative definite in the intervals [ [u], [u] + 12 ], [ [u] + 1 2 , [u] + 1 ] respectively. The integral in the interval [ [λn ], [λn ] + 1 ] is positive, negative or zero depending on the location of λn . One can always choose the value of λn to obey the relations in eq-(2.31). To sum up, given a set of N + 1 integers n = 0, 1, 2, 3, ......N bounded by N , Eqs-(2.30, 2.31) yield a system of N + 1 equations which in principle determine the values of the N + 1 unknown cut-off parameters λo , λ1 , λ2 , ....... , λN in terms of the ordinates of the zeta zeros E1 , E2 , E3 , ....., EN . The value of λo has already been fixed from eq-(2.30). Eqs-(2.31) determine the remaining ones λn , n = 1, 2, 3, ....N . Finally, after solving eqs-(2.30, 2.31), the defining relations 18

Λ(Vo ; N ) =

k=N X

ak (Vo )k = λo ⇒ Λ(Vo = 0 ; N ) = ao = λo .

(2.32)

k=0

and Λ(V = En ; N ) =

k=N X

ak (En )k = λn .

(2.33)

k=0

yield a linear system of algebraic equations for the N coefficients a1 , a2 , .... , aN associated with the truncating series (2.29), and whose solutions are given in terms of the imaginary parts of the zeta zeroes En and the values of the cutoff parameters Λ(Vo = 0) = λo , Λ(En ) = λn . The latter cut-off parameters have themselves been determined from the solutions to eqs-(2.27, 2.28). The solutions for the coefficients ak can be written compactly in terms of the van der Monde determinant ∆ of the (N + 1) × (N + 1) matrix comprised of N rows whose entries in the n-th row are 1, En , En2 , En3 , ........, EnN , for n = 1, 2, 3, ...., N . The first row has entries 1, 0, 0, 0, ....., 0 since Vo = Eo = 0, so the total number of rows and columns is N + 1. The van der Monde determinant is Y ∆ = (Ei − Ej ), f or i > j. (2.34) The other determinants involved in the solutions ∆k correspond to the (N +1)× (N + 1) matrices obtained by replacing the k-th column by a column comprised of the entries λo , λ1 , λ2 , λ3 , ......., λn . The solutions for the coefficients that define the cut-off function Λ(E, N ) at level N are compactly written as

(N )

ak

=

∆k ( λo , {λn } ; Vo = 0; {En } ) Q . i > j, i, j = 0, 1, 2, 3, .......N. (2.35) (Ei − Ej )

with Eo = Vo = 0. If the large N limit converges (N )

limN →∞ ak ( λo , λ1 , λ2 , ...., λN ; E1 , E2 , ....., EN ) → a∗k , a f ixed point. (2.36) to a fixed point, the full fledged energy-dependent cut-off function Λ(E) is determined by the infinite series Λ(E) =

k=∞ X

a∗k E k .

(2.37)

k=0

which is defined in terms of the infinite number of coefficients given by the infinite number of f ixed points a∗k . The spectral statistics of Random Matrix Models in the large N limit have been known for a long time to have deep connection to the zeta zeroes since Montgomery-Dyson found the pair-correlation functions of the ordinates of the zeta zeroes with normalized spacings in terms

19

2 of an integrand involving the function 1−( sinπx πx ) . This function is the pair correlation function for the eigenvalues of very large Random Hermitian matrices measured with a Gaussian measure (the Gaussian Unitary Ensemble) [21]. Since fixed points in Renormalization Group (RG) techniques in QFT are ubiquitous, it is warranted to explore the connections among the putative fixed points a∗k with the fixed points associated with the beta function in QFT. A RG analysis was performed by Peterman [23] to shed some light as to why the density of primes numbers decreases as 1/logx. The zeta function has also been used extensively in Regularization methods (of infinities) in QFT, see [24] and references therein. The ”Russian doll Renormalization” group has been found to have connections to the RH [19]. Finally, once the cut-off function Λ(V ) is constructed from the definition

Λ(V ) =

k=∞ X

a∗k ( λo , {λn }; Vo = 0; {En } ) V k .

(2.38)

k=0

for all values n = 1, 2, 3, ...., ∞, the sought-after potential is implicitly determined from the fundamental result 2x(V ) d N (V ; Λ(V )) 2x = = . (2.39) π π dV where the functional form of N (V ; Λ(V )) is given by the Riemann-von Mangoldt formula (2.12) by replacing E → V and by inserting the energy-dependent cut-off Λ(V ) found in eq-(2.38) into the upper limit of the integral appearing in the definition of the δ(E) terms in eq-(2.16). Naturally, due to numerical limitations, the potential can only be constructed iteratively, level by level, N, N + 1, N + 2, ......, ∞. Another salient feature to look for is to verify that (N ) the family of coefficients ak does indeed converge to the fixed points values a∗k when N → ∞. This is where the results of large N Random Matrices methods are relevant. The second case scenario is more complicated to solve if one forces Vo to lie in the domain 1 ≤ Vo < E1 . Choosing a particular value of Vo = Vo∗ in that range, one has for eqs- (2.29) Λ(Vo∗ ; N ) =

k=N X

ak (Vo∗ )k = λ∗o .

(2.40)

k=0

and Λ(V = En ; N ) =

k=N X

ak (En )k = λn .

(2.41)

k=0

when Vo = Vo∗ 6= 0, one has new solutions for the coefficients ao , a1 , a2 , ......aN ak = ak (λ∗o , λn ; Vo∗ , En ) =

∆k ( λ∗o , {λn } ; Vo∗ , {En } ) Q ; (Ei − Ej )

20

i>j

(2.42)

for i, j = 0, 1, 2, 3, .......N and where the first row of the matrix involved in the van der Monde determinant now must include the 1, Vo∗ , (Vo∗ )2 , ......., (Vo∗ )N terms. Equipped with the above solutions ak (2.42) one inserts them into the expression for the derivative term

(

k=N X dΛ(V, N ) ) (Vo∗ ) = ak ( λ∗o , λn ; Vo∗ , En ) k (Vo∗ )k−1 . dV

(2.43)

k=0

which appears in eq-(2.27). The latter equations (2.27, 2.40, 2.41, 2.42, 2.43) combined with eq-(2.28), where now N (Vo∗ , λ∗o ) 6= 0, furnishes a system of N + 1 equations which determines in principle the numerical values for the cut-off parameters λ∗o , λn ’s . From these latter values one reconstructs the the ak coefficients from eqs-(2.42), and which in turn, will determine the sought-after cut-off function Λ(V ; N ) eq-( 2.29) (at level N ), so that finally one can write the full explict form of N (V, Λ(V )) (given by the Riemann-van Mongoldt formula). )) = (2x(V )/π) which finally furnishes the form of Its derivative yields dN (V,Λ(V dV the sought-after potential V (x) (implicitly). Naturally, the equations to solve in this second case scenario are far more difficult that the ones when one simply chooses Vo = 0 simplifying drastically these calculations. There is a more general third case scenario when one has N + 2 undetermined parameters Vo∗∗ , λ∗∗ o , λn , n = 1, 2, 3, ...., N at each level N which are constrained to obey N + 2 equations given by the N + 1 eqs-(2.27, 2.28), plus an additional extra condition involving the second derivatives N 00 (Vo∗∗ , λ∗∗ o ) = 0. The procedure to solve this far more complicated system of N + 2 equations is still similar to the second case scenario, the only difference is that now Vo∗∗ is not put in by hand, but instead is another unknown parameter to be determined from the solutions of these N +2 equations. The open question remains whether or not the solutions for Vo∗∗ fall in the range 0 ≤ Vo∗∗ < E1 . Even perhaps, the solution for Vo∗∗ might be negative. Out of these three cases, the simplest one to follow is the one when one takes Vo = 0 which simplifies drastically all the calculations and allows us to provide with actual numerical results. The full numerical analysis of eqs-(2.27, 2.28, 2.29, ...) is beyond the scope of this work. It requires very sophisticated computations. To complete this subsection, we need to discuss the nature of the solutions Ψ(x) to the one-dim Dirac-like equation (2.1) on the line [−∞, ∞] given by Z x ΨE (x) = Ψo,E exp [ i P (x0 ) dx0 ] = −∞

Z

x

( E − V (x0 ) ) dx0 ].

Ψo,E exp [ i

(2.44)

−∞

where E is a real-valued continuous parameter and Ψo,E is a constant amplitude.

21

The operator −i∂/∂x would be self-adjoint in the full line [−∞, ∞], or in the compact interval [xa , xb ], if one could impose suitable vanishing boundary conditions on the above ΨE (x) solutions at ±∞ and/or at xa , xb . However, the solutions ΨE (x) are just proportional to a pure phase factor so the Ψ’s are nonvanishing for all value of x, unless one constrains the amplitudes Ψo,E to zero that will render the solutions trivial. Therefore, since one cannot find nontrivial solutions ΨE (x) obeying the boundary conditions ΨE (x = ±∞) = 0 and/or ΨE (x) = 0 at xa , xb , the operator −i∂/∂x is not self-adjoint in [−∞, ∞], or in the compact interval [xa , xb ], for the space of solutions given by (2.44). If one had h2 2 2 a second order operator D2 = −¯ 2m (∂ /∂x ) like the Schroedinger operator, there are no problems with finding self-adjoint extensions. For example, a free particle inside a box of length 2L admits normalizable wave-functions Ψn (x) ∼ sin ( nπx L ) obeying the suitable boundary conditions Ψn (x = ±L) = 0. For these reasons we conclude that the self-adjointness property is not required to fulfill our goals. We saw in section 1 how by working with a pair of non-Hermitian Hamiltonian operators was sufficient to show why it was the CT symmetry which f orced the energy spectrum to be real : Es = s(1 − s) = Es∗ = s∗ (1 − s∗ ), leading to the only possible solutions s = 21 + iρ and/or s = real, and consistent with the fact that the Riemann zeta function has trivial zeroes on the negative even integers and nontrivial zeroes in the critical line Re s = 12 . For the Dirac-like operator (2.1) all we need is to impose P T symmetry where this time by T reversal symmetry we do not mean inversion t → (1/t) ⇒ log(t) → −log(t), but the standard t → −t symmetry used in P T symmetric QM. The momentum p = dx/dt is invariant under P T symmetry since x and t both reverse signs, this means that i → −i under P T symmetry so that the momentum operator remains invariant pˆ = −i¯h(∂/∂x). There is nothing strange by having i change sign under P T symmetry since Clifford algebras in D = 1 have two generators, the identity element 1 and the 1 × 1 matrix γ whose entry is just −i, so that {γ, γ} = 2i2 = −21, if one takes the metric of the one-dim space to be g11 = −1. Therefore under P T symmetry γ → −γ which implies that i → −i. This leaves us with having to impose the condition V (−x) = V (x) on the potential in order to have a P T symmetric Dirac-like operator −i∂/∂x + V (x). In order to define V (x) in the regions x < 0 one must choose the minus sign in )) front x(V ) = − π2 dN (V,Λ(V . The positive sign selects the solutions in the region dV x > 0. Furthermore, it is important to emphasize that the one-dimensional Jacobian (from the change of variables) is (dx/dV ) in the x > 0 region, but it is −(dx/dV ) in the mirror x < 0 region. There is a crucial sign change to ensure that the portion of the line-integral along the left region does not trivially cancel out the portion of the line-integral in the right region. The Bohr-Sommerfeld quantization rule involves theH closed contour in phase space that in the case of R∞ a symmetric potential gives pdx = 4 o pdx = 2nπ, thus care must be taken with the signs of dx/dV . To sum up this discussion : the self-adjointness (Hermitian) property is not

22

required to prove the Riemann Hypothesis. What matters was the CT symmetry in section 1 and P T symmetry in this section related to the spectrum of the Dirac-like operator in one-dimension. To finalize, once we extend the domain of V (x) to the region x < 0 by taking the mirror image of the potential constructed in this section; the solutions associated with the discrete family of zeta zeroes En (embedded in the continuum of solutions) are is simply obtained by inserting the value of E = En inside the integrand of (2.45) Z x 1 ( En − V (x0 ) ) dx0 ]. ΨEn (x) = √ exp [ i (2.45) L −∞ where in the region x < 0 one must use the branch of the potential solution )) to ensure that indeed we are selecting solutions given by x(V ) = − π2 dN (V,Λ(V dV which obey V (−x) = V (x). L is an inf rared cutoff that is required so that the wave-functions ΨEn (x) are square integrable on the line Z

L/2

limL→∞ { −L/2

Ψ†En (x) ΨEn (x) dx } = 1.

(2.46)

One must take the L → ∞ limit af ter the integration (2.47) is performed and not before otherwise one gets the trivial result for the wavefunctions Ψ = 0. Finally, when one evaluates the discrete family of wave functions at the cusps points xn = +∞ (where the values of the potential are V (x = xn = +∞) = En ) one arrives at Z xn =∞ 1 ( En − V (x0 ) ) dx0 ] = ΨEn (x = xn = ∞) = √ exp [ i L −∞ Z xn =∞ 1 √ exp [ 2 i ( En − V (x0 ) ) dx0 ] = L 0 1 √ exp { i π [ N (En ; Λ(En )) − N (Vo ; Λ(Vo )) ] } = L 1 (−1)n √ exp [ i n π] = √ . L L

(2.47)

as a direct result of the conditions in eq-(2.28) and eq-(2.3). Therefore, at the cusps points x = xn = +∞ the wave functions ΨEn (xn ) alternate in sign. This changing of sign is related to the presence of Gram points in the Riemann Siegel formula, with the only difference that the phase factors in (2.47) involve the full-fledged zeroes (discrete energy levels En ) counting function N (E, Λ(E)) whereas in the Riemann-Siegel formula only the average energy level counting function is used given by the first two terms of eq-(2.12) [2]. The values of the wave-functions at the x = −∞ are simply ΨEn (x = −∞) = √1 . For even n the wavefunctions are periodic (with an inf inite period) in L the sense that ΨEn (x = −∞) = ΨEn (x = +∞). For odd values of n the wavefunctions are anti-periodic in the sense ΨEn (x = −∞) = − ΨEn (x = +∞). 23

Therefore, the imaginary parts of the zeta zeroes En in the critical line are the only values among the E-continuum of values which obey the boundary conditions ΨE (x = −∞) = ± ΨE (x = +∞). This physical interpretation of the discrete values En among the E-continuum must have bearing on the periodic orbits associated with the ”chaotic” Riemann dynamics whose periods are multiples of logarithms of prime numbers as described by Berry and Keating, and based on their classical Hamiltonian H = xp [14] Notice that if one were to replace the values λn and Λ(Vo ) for Λ = ∞, one would have for phases in the wave functions (2.47) the following values π { N (En ; Λ = ∞) − N (Vo ; Λ = ∞) } = π { n − δn − N (Vo ; Λ = ∞) } = π n − φn − φo .

(2.48)

where now the zero-point energy Vo is the one determined from the solution to the transcendental equation (2.26). Therefore, if no cut-offs are set in the upper limits of the integral defining the δ(E) terms (2.16, 2.17) there would be a nontrivial phase − shif t in the wave functions as shown in eq-(2.48) and one would no longer have the nice periodicity (anti-periodicity) behaviour as before; i.e. one would have now a quasi-periodicity behaviour. To finalize this section we return to the solution of Abel’s integral equation (2.11) (α = 21 ) in the Schroedinger operator case x(V ) − x(Vo ) =

1 Γ(1/2)

Z

V

0

1 dN (E) p dE. dE (V − E)

(2.49)

To control the divergences in the integral once again we may introduce a cut-off function Λ(E) in the counting function and have dN (E,Λ(E)) inserted into the dE above integral where the cut-off function Λ(E) is defined by the series expansion of eq-(2.38). The coefficients a∗k are the f ixed points of the large N limit of (N ) the family ak (λo , Vo = 0; {λn }, {En }) given explicitly by the relations (2.35) involving the van der Monde determinant. The values of the λn ’s as functions of En are obtained by solving eqs-(2.28), however, now the value of λo is no longer determined from eq-(2.27), because that equation no longer applies, but it is left out as a free parameter that is related to the integration constant x(Vo = 0) = xo in the l.h.s of (2.48). Therefore, one has now, at each level N, N + 1, N + 2..... a well behaved cutoff function Λ(E, N ) in the expression N (E, Λ(E)) given by eq-(2.12) that can be inserted into the above integral (2.48), and provide solutions for x(V ) − xo ( x(Vo = 0) = xo ), and which defines implicitly, the potential V (x) of the selfadjoint Schroedinger operator defined in the whole real line that reproduces the zeta zeros.

24

3

Area Quantization in Phase Space, Duality, Spacetime Singularities, Renormalization Group and Distribution of Primes and Zeta Zeros

To finalize this work , we will derive the Area Quantization condition in Phase Space An = n π of the intervals [0, En ] for n = 1, 2, 3, ......∞ and show why area quantization is one physical reason why the average distribution of primes density for very large x given by O( log1 x ), has a one-to-one correspondence with the inverse average density of zeta zeros in the critical line. As the number density of primes decreases asymptotically with large x as (1/log x), the average density of zeta zeros in the critical line increases asymptotically (for very large 1 E E) as 2π log( 2π ). This finding is consistent with the results of Petermann [23] who found the 1/logx behaviour to be connected to the Renormalization Group program in QFT. In the previous section we found that the potential function V (x) obtained implicitly from eq-(2.40) turns out to be a multi-valued function of x which requires splitting the energy regions into different bands, branches, like a nonperiodic crystal lattice [0, E1 ], [E1 , E2 ], [E2 , E3 ], ............ , [En−1 , En ], ......

(3.1)

such that at the boundaries of those bands : x(V = En ) = π2 N 0 (V = En ) = ∞ due to the discontinuity of the S(E) term (2.12) at En . The left and right derivatives of x(V ) at V = En are (dx/dV ) = ±∞ which is also consistent with taking the second derivatives of the Heaviside step function Θ00 (E − En ) = δ 0 (E − En ), since the counting function is defined by PN N (EN ) = 1 Θ(E − En ). In the infinitesimal region V = En ± n , for a suitable infinitesimal n (En ) > 0, one expects a sudden jump of the function N (V, Λ(V )), from values less than n, to values greater than n, while reaching the precise value of n at V = En due to the conditions imposed in (2.28). This sudden jump is provided for by the S(E) term in the Riemann-von Mangoldt formula. On a separate problem, we exploited this singular behaviour of the derivatives of the Heaviside step function to construct a different solution to the static spherically symmetric gravitational field produced by a point mass M at r = 0 than the standard text book solution. The solutions for the metric [32] were continuous except at the location r = 0 of the point mass, leading to a delta function for the scalar curvature R = (2GM δ(r)/r2 ) instead of R = 0. The Euclideanized Einstein-Hilbert action coincided precisely with the Black Hole Entropy where the area of the horizon which has now been displaced at the location r = 0+ (due to the discontinuity of the metric at r = 0 ) is the usual value 4π(2GM )2 . The area-radial function chosen was ρ(r) = r + 2G|M |Θ(r) so that ρ(r = 0) = 0; ρ(r = 0+ ) = 2G|M |; ρ(r = 0− ) = −2G|M | due to the definition of the Heaviside step function : it is 1 for r > 0; −1 for r < 0, and

25

is 0 for r = 0. This discontinuity has the same form as the discontinuity of the argument of ζ( 12 + iE). For this reason we believe that John Nash’s approach to the RH based on spacetime singularities was on the right path. Between two consecutive cusps where the coordinate function blows up x(V ) = π2 N 0 (V ; Λ(V )) = ∞ at V = En , En+1 , lies a ”valley” region where there (n) are inf lection points of N (V, Λ(V )) at the locations E∗ , within the intervals (n) (n) (n) 00 En < E∗ < En+1 , such that N (E∗ , Λ(E∗ )) = 0; i.e. (dx/dV ) = 0 at the (n) bottom of the valleys V = E∗ , while (dx/dV ) = ±∞ at the cusps En . One can visualize the coordinate graph function x(V ) as an infinitely long suspension bridge (from V = 0 to V = ∞) with infinitely high poles/spikes x(V ) = ∞ at the specific locations V = E1 , E2 , E3 , ......., and with the suspension cables falling into the U -shaped valley regions in between. With this picture in mind, the areas An in the Phase Space comprised by this non − periodic crystal lattice of peaks and valleys, are quantized in multiples of π as follows xn =∞

Z 2

Z

0

Z

(En − V ) 0

E3

(En − V )

2 E2 (n)

A1

(En − V ) 0

E1

2 Z

En

P (x) dx = 2

(n)

+ A2

dx dV + 2 dV

Z

E2

(En − V ) E1

dx dV + ........... + 2 dV (n)

+ A3

dx dV = dV

Z

dx dV + dV

En

(En − V ) En−1

+ ....... + A(n) = n π; n

dx dV = dV

n = 1, 2, 3, ....

(3.2)

For a given value of n = 1, 2, 3, ..... the sum of each one of these n- aperiodiccrystal-like bands contributes to a net value of area An = n π. This is not to say that the areas in (3.2) are equally partitioned in one unit of π ! It is the whole sum which adds up to nπ. For any given value of n one can take the ratios of areas to obtain a sequence of fractions (n)

(n)

(n)

(n)

A1 A A An , 2 , 3 , ....... ; n = 1, 2, 3, ..... (3.3) nπ nπ nπ nπ (one should take the magnitude of the areas in the case of negative contributions in the integrals). It is known that the self-similarity of the Farey sequence of fractions posses remarkable f ractal properties [27] that is very relevant to the validity of the Riemann Hypothesis (RH) based on Farey fractions and the Franel-Landau shifts [28]. Do the area-fractions (3.3) follow a Farey sequence when 2x(V )/π = N 0 (V, Λ(V ))? A fractal SUSY QM model to fit the spectrum of the imaginary parts of the zeta zeros ρn was studied in [16] based on a Hamiltonian operator that admits a factorization into two factors involving fractional derivative operators whose fractional (irrational) order is one-half of the fractal dimension (d = 1.5) of the fractal potential found by Wu-Sprung [13]. A model of f ractional spin has been

26

constructed by Wellington da Cruz [29] in connection to the Fractional Quantum Hall effect based on the filling factors associated with the Farey fractions. This approach based an a f ractional Quantum Hall should be contrasted with the ordinary Quantum Hall Effect approach to the RH [19]. The integral depicting the phase space area of a domain D can be written as a result of Stokes theorem as I Z 1 (P dX − XdP ). (3.4) dX ∧ dP = 2 C D where the line (contour) integral is defined along the boundary region of the domain D of phase space. Since the potential is symmetric V (−x) = V (x) another way of obtaining the same result for the net areas is to compute the areas from the line integral (3.4) using the equality due to the symmetry of the potential I I 1 1 P dX = − X dP. (3.5) 2 C 2 The relation P + V = En ⇒ P = En − V yields I An = −

I

Z

X dP = −

En

X d (En − V ) = 2

X dV = 0

Z

En

dN (V, Λ(V )) dV = π [ N (En , Λ(En )) − N (V = 0, Λ(V = 0)) ] = n π. dV 0 (3.6) Therefore, in general, one has the dual or reciprocal forms for the same phase space area An as a direct consequence of Stokes theorem π

En

Z An = 2

Z

E1

X(V ) dV = 2 0

Z

E3

Z

En

X(V ) dV + ........... + 2 (n)

+ I2

X(V ) dV + E1

E2 (n)

E2

X(V ) dV + 2 0

2 I1

Z

X(V ) dV = En−1

(n)

+ I3

+ ....... + In(n) = n π;

n = 1, 2, 3, ....

(3.7)

)) where we have re-written x as X. Because of the relationship 2X(V ) = π dN (V,Λ(V dV derived in the previous section, and by setting N (V = 0, Λ(V = 0)) = 0, one has

Z An = 2

En

Z X(V ) dV = π

0

0

En

dN (V, Λ(V )) dV = π N (En , Λ(En )) = n π . dV (3.8)

27

From eqs-(3.7, 3,8) one can write the values of the integrals associated to each one of the respective intervals [0, E1 ], [E1 , E2 ], [E2 , E3 ], ......, [En−1 , En ] as π + π(2−1) + π(3−2) + ....... + π(n−(n−1)) = π + π + π + ....... + π = n π. (3.9) which is a direct consequence of the quantization conditions (2.28) when N (V = 0, Λ(V = 0)) = 0 N (E1 , Λ(E1 )) = 1; N (E2 , Λ(E2 )) = 2; ........., N (En , Λ(En )) = n. (3.10) R Therefore, from this decomposition of the areas in terms of X dV integrals, one has now an equipartition of the area An = nπ into n-single bits and whose quantum of area is π (n)

I1

(n)

= π, I2

(n)

= π, I3

= π, ....... In(n) = π.

(3.11)

A full cycle requires starting at −∞, going to +∞ and back to −∞, thus the full cycle will generate 2nπ area-bits, consistent why the n-th-winding number of the orbit associated with the n-th zeta zero En . This is where one can make contact with the work by Berry and Keating [14] on the periodic orbits associated with the ”chaotic” Riemann dynamics whose periods are multiples of logarithms of prime numbers based on the classical Hamiltonian H = xp [14] and the Gutzwiller trace formula. Now we are ready to find the relationship between area quantization, and the distribution of primes and zeta zeros. The following integrals Yn , for n = V 1, 2, 3.... also give the same values of nπ, after the renaming of variables y = 2πn Z 2πn Z 2πn 1 V V V ) dV = − n π ) d( ) = log ( log ( 2 0 2πn 2πn 2πn 0 Z 1 −n π log (y) dy = { − n π y ( log y − 1 ) }10 = n π. (3.12)

Yn = −

0

because 0 log 0 → 0. Upon equating the 3 integrals (3.2, 3.7, 3.12), and after using the results of the previous section x(V ) = π2 N 0 (V ) ⇒ x0 (V ) = π2 N 00 (V ), the area quantization in phase space reads An = π Z 2πn

−

1 2π

0

Z En d2 N dN (V, Λ(V )) dV = dV = dV 2 dV V =0 0 Z 1 V log ( ) dV = −n log (y) dy = n; n = 1, 2, 3, ...... (3.13) 2πn 0 Z

En

( En − V )

The fact that the integrals are equal does not mean the integrands are equal, nevertheless one can still establish the following one-to-one correspondence of the integrands and domains of integration as follows 28

−

1 V log ( ) ↔ 2π 2πn

dN (V, Λ(V )) ≡ ρ(V ); dV

2 π n ↔ En .

(3.14)

dN (V, Λ(V )) d2 N (V ; Λ(V )) . (3.15) ↔ ( En − V ) dV dV 2 From the correspondence (3.14) one learns that the irregularly spaced zeta zeroes En has a correspondence with the evenly spaced energy levels given by V 1 1 log( 2πn ) = − 2π log(y), which has 2πn. While the logarithmic integrand − 2π the same functional form as the inverse average density of primes log(x) (up to a sign and numerical factor) has a correspondence to the density of zeta zeros V 1 log( 2πn ) has a connection to ρ(V ) in the critical line. The negative sign − 2π Connes work on the RH and Noncommutative Trace formula where the location of the zeta zeroes were interpreted as absorption lines of the spectrum, instead of emission lines [18] . The prime number theorem states the the number of primes P(N ) in the interval [0, N ], for large N , is of the order of P(N ) ∼ (N/logN ). The average number density of primes is P(N )/N ∼ (1/logN ), so its inverse is log(N ). The density of primes is instead dP(N )/dN = (1/logN ) − (1/logN )2 . We believe this is no coincidence for the even harmonious spacing of the energy levels 2πn is related to the imaginary parts of the zeros of : sin(iz) = i sinh(z) = i sinh(x+iy) = i sinh(x) cos(y) −cosh(x) sin(y) = 0 ⇒ sinh(x) cos(y) = 0, and cosh(x) sin(y) = 0.

(3.16)

The solutions to these last 2 equations is x = 0, y = ±2πn. Therefore the zeroes of the function sin(iz) = i sinh(z) = 0 are are zn = 0 ± i2πn, which satisfy an analog of the Riemann Hypothesis. They all line in the vertical line Re(z) = 0, with the main difference being that the latter zeros are all evenly and harmoniously spaced in intervals of 2π along the imaginary axis. Thus, the En ↔ 2πn correspondence would be another reflection of the irregular but ”harmonious” distribution of the primes. It is warranted to explore the connections to the area quantization of the quantum droplets in the Quantum Hall Effect. Studies of the Lowest Landau Levels in the quantum mechanical model for a charged particle on a plane in a constant uniform perpendicular magnetic field by [19], have shown to yield the absorption level spectrum of the zeta zeros by Connes [18] and related to the Berry-Keating [14] model of the average level density of the zeta zeroes based on the classical Hamiltonian H = xp. It was conjectured [19] that the f luctuating part S(E) of the counting function N (E) (2.12) might be accounted for by the higher Landau levels. The upshot of our results is that we have been involved with the full-fledged Riemann-van Mangoldt expression N (E) in eq(2.12) which not only has the fluctuating part S(E), but also the higher order

29

O(E −n ) corrections as well, the δ(E) terms, in addition to the standard terms E E 2π (log 2π − 1) + 7/8. To finalize, we should add that since we are dealing with Dirac-like operators one must not forget the existence of anti-particles with negative energy states −En , although we are not working in four dimensions where the CP T theorem applies. Negative energy states is consistent with the fact that the zeta zeros in the critical line appear in pairs of complex conjugates 21 ± iEn . The absence of a positive-energy electron behaves as if a positron of positive charge and negative energy were created. Eq-(2.1) admits the analog of negative energy in onedimension, ( or 0 + 1-dim ) by simply writing the dispersion relation (P + V )2 = M2 ⇒ (P + V ) = ±M = ±E. Therefore, the existence of ±E eigenvalues is compatible with the zeros appearing in pairs of complex conjugates 12 ± iEn along the critical line. In Electro-Magnetism (EM) the canonical momentum is defined by the replacement pµ → pµ −eAµ , where Aµ is the EM potential and −e is the negative charge of the electron. Thus having the generalized momentum P + V bears some relation to the canonical momentum in EM , which brings up again the connection to the work on Landau Lowest Levels, Quantum Hall Effect... by [19]. In future work we will explore the relations of our work to • Chaotic Renormalization Group Flows, Universal Mandelbrot Set, Phase transitions, attractors, Julia sets, .... by [31]. • Fractal strings, fractal membranes, noncommutative spaces, Dirac-like operators, spectral triples , quasi-crystals, modular flows of the moduli space of fractal membranes, adeles, arithmetic geometries, ..... in connection to the flows of zeroes of zeta functions towards the critical line, by Lapidus et al [26] • Connes noncommutative trace formula [18]. The fermionic Trace Formula, supersymmetry, Witten index and the Mobius function [30] • Black Hole entropy and area quantization in Loop Quantum Gravity; Farey sequences, fractal statistics, and the fractional Quantum Hall Effect [29]. • Cyclotomy, Phase quantization, Ramanujan sums, ...... by Planat et al [25]. To summarize this work, in sections 1 and 2 we have presented two plausible methods to prove the Riemann Hypothesis. One was based on the modular properties of Θ functions and the other on the Hilbert-Polya proposal to find an operator whose spectrum reproduces the ordinates of the zeta zeros in the critical line. We described in detail how the Dirac-like operator with a potential V (x) in eq-(2.1) reproduces the spectrum after the boundary conditions ΨE (x = −∞) = ± ΨE (x = +∞) are imposed. Such potential V (x) was derived implicitly from the relation x = x(V ) = π2 ρ(V ) = π2 (dN (V, Λ(V ))/dV ). At the end of section 2 , we explained how to provide the implicit form of the potential V (x) for the self-adjoint Schroedinger operator that also reproduces the zeta zeros. Crucial in the construction, was the introduction of an energydependent cut-off function Λ(E). In the final section 3 the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr-Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning as to why 30

1 the average density of the primes distribution for very large x : O( logx ), has a one-to-one correspondence with the asymptotic limit of the inverse average density of zeta zeros in the critical line. Acknowledgements

We thank Frank (Tony) Smith and Jorge Mahecha for discussions and help with Mathematica; to Paul Slater for bringing to my attention the work of A. Karatsuba and M. Korolev, and M. Bowers for assistance.

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